Washington (CNN)Senate Republicans are becoming increasingly frustrated with Scott Pruitt, the administrator of the Environmental Protection Agency, as he faces an array of ethical questions ahead of scheduled appearances on Capitol Hill this week.

While many Democrats have already called on Pruitt to resign, Republicans in Congress have largely held their fire, taking a wait-and-see approach. But as reports continue to raise issues — from questions over renting a condo from a lobbyist friend to a penchant for 24-hour security and first-class travel — Republican lawmakers are growing impatient.

"They are not good reports. He needs to answer those questions," said Sen. John Thune, the No. 3 Republican in the Senate. "There's a lot stuff out there around him that is certainly not helpful to his or the administration's cause."

The inspector general at the EPA has four open investigations involving Pruitt, while the Republican-led House Oversight Committee is also looking into his actions.

Republican Sen. John Kennedy of Louisiana said "some of his behavior has hurt the President of the United States. It's hurt the President's credibility. It has hurt the credibility of all of us. And it would be way cooler if he'd behave."

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Kennedy stopped short of calling for Pruitt to resign, saying he hopes Pruitt has learned a lesson. "You can't just go around acting like a big shot," he said.

Sen. John Barrasso, chairman of the Senate Environment and Public Works Committee, said he "absolutely" has concerns about the growing list of controversies. "I've been asking the questions -- they still need to be answered," he said.

Republican Sen. Shelley Moore Capito of West Virginia said she'll be speaking with Pruitt one on one later this week. "I think they're perfectly valid questions and they need to have good answers."

Meanwhile, senior White House officials are debating whether the embattled EPA administrator can continue in his role, sources familiar with the matter told CNN. Pruitt still has allies in the administration urging Trump to keep him on board, but some aides to the President question whether Pruitt simply has the ethics necessary for government service, according to one White House source.

Senate Majority Leader Mitch McConnell described himself as a "supporter" of Pruitt at a news conference on Tuesday, but declined to say whether he has concerns about his actions, saying, "We'll just see."

"I think he's going to make an appearance up here ... this week," he continued. "I expect there will be a lot of interest."

Indeed, all eyes will be on Congress Thursday when Pruitt appears before the House Appropriations Committee and the House Energy and Commerce Committee. While he's set to testify on the EPA budget, it's widely expected he'll get a grilling on his scandal-ridden tenure as the agency's chief.

"There will be questions, I'm sure, related to him more than appropriations," said Republican Sen. James Lankford, who represents Pruitt's home state of Oklahoma.

A White House official said Pruitt refused multiple offers of assistance to prepare for the upcoming hearings, a decision that has baffled aides. While Trump appears to continue to support Pruitt, the administrator's performance this week could change that. "Let's see what happens," the official said.

Lankford said he's waiting for internal investigations to play out at the EPA before he forms an opinion on Pruitt's future, and he invoked the name of the former Health and Human Services secretary who was fired by President Donald Trump last year.

"The President's made decisions on people before, obviously with Tom Price. It went through the process, did the evaluation, made a decision," he said. "I'll let the administration continue to go through that now."

Republican Sen. John Boozman of Arkansas, who sits on the Environment and Public Works Committee, expressed confidence that Pruitt would also appear on the Senate side in the "not-too-distant future."

"We're in a situation where various things are being raised," he said. "At some point, we need to follow up and make sure it's dealt with."

Republican Sen. James Inhofe, the senior senator from Oklahoma, has close ties with the EPA as nearly a dozen of his former staffers now work in senior positions at the agency. While he has previously defended Pruitt — whom he said Tuesday he's known "for a long time" -- he's taken a less ardent tone in recent days.

"There are people out there that will embellish in whatever way will serve their purposes to get rid of this guy and he's accomplished a lot," Inhofe told CNN. "But I do want to check out all of the allegations against him."

Part of the waning GOP support for Pruitt may stem from Andrew Wheeler's confirmation earlier this month as deputy EPA administrator, someone who sources describe as a seasoned and known quantity for lawmakers.

"That makes it a bit easier for Republicans to part ways with Pruitt now that Wheeler is confirmed," said one source with ties to Inhofe.

Some Republicans were less than eager to see congressional involvement in the controversy. Sen. John Cornyn, the No. 2 ranked Republican in the Senate, sought to put some distance between Congress and Pruitt on Tuesday when asked whether the administrator needs to fight it out and stand firm. "You know our job is really done when we confirmed Mr. Pruitt," he said. "This is solely up to the President."

Cornyn, when asked whether the Senate has an oversight role over Cabinet secretaries, pointed to efforts lead by House Oversight Chairman Trey Gowdy in investigating the concerns.

"I'm not opposed to us doing oversight but we have a lot to do here -- no need to duplicate efforts," Cornyn said.

Republican Sen. Susan Collins of Maine argued the controversies have become a major distraction.

"We're seeing a constant drip-drip-drip of ethical issues that are being raised against him," she said. "But it's up to the President, not to me, whether or not he stays in that position."